I've searched online, but I have not found any clear instructions how to permanently add .sty files to my TeXShop installation. I'm using a Mac with OSX (TeXShop is a mac only program).

What is the best way of doing this? The one comprehensive set of instructions I found here.

I've got as far as extracting the files (running LaTeX on the .ins and . dtx files) but after that I don't know what to do. I also don't understand how to update the index of my distribution, and whether there are particular OSX specifics (the instructions are for Windows).

2 Answers
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The first question you should ask is whether the package you are trying to install is already installed on your system. If you are using MacTeX as your distribution (which you probably are) then most packages are already available. Within TeXShop you can choose the "Show Help for Package" item from the Help menu and enter your package name. (Or make a simple document that loads the package to see if it doesn't exist.)

On a Mac, local additions go into the Library/texmf folder of your Home folder. (~/Library/texmf) You never need to update the file database for local additions to this folder. If the folder doesn't exist, you will need to create it yourself. For a one-click version of this you can use this utility: Make Local texmf Folder (Disclaimer: I am the author of this).

Note for OS 10.7 or 10.8 (Lion/Mountain Lion)

If you are running OS 10.7 (Lion) or later, your user Library is unfortunately a hidden folder. It is, however, easy to access it through the Finder's Go menu: hold down the Option key while choosing the Go menu, and the local Library folder will appear in the list. Alternatively if you type Command-Shift-G you can enter the folder path directly (~/Library/texmf)

If you’re using BasicTeX (86 MB) instead of full-blown MacTeX (2.3 GB), e.g. to save valuable disk space on a MacBook Air, you’ll only find the most common packages and classes in your installation. To add more, the simplest option is the TeX Live Utility which you should find in your application folder. It also updates packages to the most recent version found on CTAN.

My /usr/local/texlive folder structure currently weighs in at well below 400 MB.